Spaces over BO are thickened manifolds

TL;DR

This paper establishes an equivalence between the infinity-category of thickened smooth manifolds and finite spaces over BO, illustrating how embedding problems become homotopy-theoretic in higher dimensions without relying on smooth approximation or h-principle.

Contribution

It formalizes the relationship between thickened manifolds and homotopy-theoretic spaces over BO, providing a new perspective on embedding questions in higher dimensions.

Findings

The infinity-category of thickened manifolds is equivalent to finite spaces over BO.

A geometric construction of pushouts in this category is developed.

The proof avoids smooth approximation and h-principle techniques.

Abstract

Consider the topologically enriched category of compact smooth manifolds (possibly with corners), with morphisms given by codimension zero smooth embeddings. Now formally identify any object X with its thickening X x [-1,1]. We prove that the resulting infinity-category of thickened smooth manifolds is equivalent to the infinity-category of finite spaces over BO. (This is one formalization of the philosophy that embedding questions become homotopy-theoretic upon passage to higher dimensions.) The central tool is a geometric construction of pushouts in this infinity-category, carried out with an eye toward proving analogous results in exact symplectic geometry. Notably, the proof never invokes smooth approximation nor any h-principle.